Real Analysis Ep 20: Heine-Borel Theorem

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Episode 20 of my videos for my undergraduate Real Analysis course at Fairfield University. This is a recording of a live class.

This episode is some more about the Heine-Borel Theorem.

Chris Staecker

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I'm a little lost on the proof that closed and bounded implies that every open cover has a finite subcover (finishes at 25:40). The passage from "By NIP, we end up with one point x" to "then this single point {x} requires infinitely many sets to cover." appears kind of unjustified to me. I mean... I kind of get why it should work, but at the same time it seems like it shouldn't? To me, it sounds akin to arguing that "since every set in [0, 1/n] has infinitely many numbers, the intersection of all these sets should have infinitely many numbers as well", which is clearly false. In both cases it looks like we're extending a property of the sets we're intersecting (those intervals in one case and the [0, 1/n] in the other) to their intersection proper, without explaining why, since we have no reason to assume that the intersections was one of the sets we began with.

jvf

Thanks for the video. I don't quite understand the proof of Heine-Borel theorem. Why the intersection of all the sets contains only single point 23:58 while Cantor set ends up with infinite points? btw, I feel the idea of open cover and finite subcover is so smart and non trivial to me. Is there any background about it? How do people discover them?

chaolin

Question: At 8:28 for a contrapositive proof what is the rule for dealing with words like “some” and “any.” Is this a rule that when you do a contrapositive proof “any” becomes “some” and vice versa? Thanks!

sanjeev

Thanks for uploading these videos..! I've only watched up until this point of the series so I don't know if you address what Im going to ask later but here it goes. Whats the relationship between the concepts of connection and disconnection and continuity and discreteness? Could you say that a connected set is equivalent to a continuous set?

juliopaniagua

Proof using more words than Greek symbols. This is yucky!

hunterroy

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